Quadratic Formula

A quadratic equation has the form:
ax² + bx + c = 0 (as long as a ≠ 0)
- you need "= 0" on one side
The quadratic formula is a formula that gives both solutions:
- $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Read off the values of a, b and c from the equation
Substitute these into the formula
- write this line of working in the exam
- Put brackets around any negative numbers being substituted in
To solve 2x² - 7x - 3 = 0 using the quadratic formula:
- a = 2, b = -7 and c = -3
- $x = \frac{- (- 7) \pm \sqrt{{(- 7)}^{2} - 4 \times 2 \times (- 3)}}{2 \times 2}$
- Type this into a calculator or simplify by hand
  - once with + for ± and once with - for ±
- The solutions are x = 3.886 and x = -0.386 (to 3 dp)
  - Rounding is often asked for in the question
  - The calculator also gives these solutions in exact form (surd form), if required
  - x = $\frac{7 + \sqrt{73}}{4}$ and x = $\frac{7 - \sqrt{73}}{4}$
- On the non-calculator paper you will be asked to give your answers as simplified surds so make sure that the number under the square root has no square factors
  - If it does then simplify the surd!

The part of the formula under the square root (b² – 4ac) is called the discriminant
The sign of this value tells you if there are 0, 1 or 2 solutions
- If b² – 4ac > 0 (positive)
  - then there are 2 different solutions
- If b² – 4ac = 0
  - then there is only 1 solution
  - sometimes called "two repeated solutions"
- If b² – 4ac < 0 (negative)
  - then there are no solutions
  - If your calculator gives you solutions with i terms in, these are "complex" and not what we are looking for
- Interestingly, if b² – 4ac is a perfect square number ( 1, 4, 9, 16, …) then the quadratic expression could have been factorised!

Make sure the quadratic equation has "= 0" on the right-hand side, otherwise it needs rearranging first
Always look for how the question wants you to leave your final answers
- for example, correct to 2 decimal places

Use the quadratic formula to find the solutions of the equation 3x² - 2x - 4 = 0.
Give each solution as an exact value in its simplest form.

Write down the values of a, b and c

a = 3, b = -2, c = -4

Substitute these values into the quadratic formula, $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
Put brackets around any negative numbers

$x = \frac{- (- 2) \pm \sqrt{{(- 2)}^{2} - 4 \times 3 \times (- 4)}}{2 \times 3}$

Simplify the expressions

$x = \frac{2 \pm \sqrt{4 + 48}}{6} = \frac{2 \pm \sqrt{52}}{6}$

Simplify the surd

$x = = \frac{2 \pm \sqrt{4 \times 13}}{6} = \frac{2 \pm 2 \sqrt{13}}{6}$

Simplify the fraction

$x = \frac{1 \pm \sqrt{13}}{3}$

Quadratic Formula (Cambridge O Level Maths)